Dynamic Window GAM Analysis: Sunset Window (Max Count → Functional Sunset)

1 Background and Rationale

This analysis addresses feedback from Francis regarding the temporal alignment of weather predictors with butterfly responses. The original daily-level analysis used fixed 6am-6pm weather windows, which Francis identified as having a temporal logic issue:

“All of the metrics for wind, temperature and light, would need to be re-calculated for 24 hour periods that begin at the time of the highest count.”

1.1 Francis’s Key Points

1. Temporal alignment: Butterflies can only respond to weather after it occurs
2. Biological timing: If max count occurred at 2pm on day t-1, the relevant weather window should start at 2pm, not 6am
3. Roosting decisions: Weather from max count through sunset determines whether butterflies abandon the roost

1.2 This Analysis

Weather window: time_of_max_count_t_1 → last_observation_time_t (functional sunset)

• Weather metrics now include overnight conditions (24/7 temperature and wind)
• Window lengths vary by observation (mean: 29.6 hours, range: 22.5-34.9 hours)
• Captures weather exposure until roosting decisions are finalized at dusk
• Tests the hypothesis: “Do weather conditions from peak count to sunset predict roost abandonment?”

Data source: data/monarch_daily_lag_analysis_sunset_window.csv (103 lag pairs)

---

2 Setup

library(tidyverse)
library(mgcv)
library(knitr)
library(here)
library(corrplot)
library(patchwork)
library(gratia)

# Load the sunset window data
daily_data <- read_csv(here("data", "monarch_daily_lag_analysis_sunset_window.csv"))

# Create the square root transformed response variable
daily_data <- daily_data %>%
    mutate(
        butterfly_diff_95th_sqrt = ifelse(butterfly_diff_95th >= 0,
            sqrt(butterfly_diff_95th),
            -sqrt(-butterfly_diff_95th)
        )
    )

3 Data Overview

cat("Dataset dimensions:", nrow(daily_data), "rows x", ncol(daily_data), "columns\n")

Dataset dimensions: 103 rows x 52 columns

cat("Number of deployments:", n_distinct(daily_data$deployment_id), "\n")

Number of deployments: 7 

cat("Date range:", min(daily_data$date_t), "to", max(daily_data$date_t), "\n\n")

Date range: 19680 to 19756 

cat("Window characteristics:\n")

Window characteristics:

cat("- Mean lag duration:", round(mean(daily_data$lag_duration_hours, na.rm = TRUE), 2), "hours\n")

- Mean lag duration: 29.56 hours

cat(
    "- Duration range:", round(min(daily_data$lag_duration_hours, na.rm = TRUE), 2), "-",
    round(max(daily_data$lag_duration_hours, na.rm = TRUE), 2), "hours\n"
)

- Duration range: 22.5 - 34.9 hours

cat("- Median data completeness:", round(median(daily_data$metrics_complete, na.rm = TRUE), 3), "\n\n")

- Median data completeness: 1 

# Summary of key variables
summary_vars <- daily_data %>%
    select(
        butterflies_95th_percentile_t,
        butterflies_95th_percentile_t_1,
        butterfly_diff_95th,
        temp_max,
        temp_min,
        temp_at_max_count_t_1,
        wind_max_gust,
        sum_butterflies_direct_sun,
        lag_duration_hours,
        metrics_complete
    )

kable(summary(summary_vars), caption = "Summary statistics for key variables")

Summary statistics for key variables

	butterflies_95th_percentile_t	butterflies_95th_percentile_t_1	butterfly_diff_95th	temp_max	temp_min	temp_at_max_count_t_1	wind_max_gust	sum_butterflies_direct_sun	lag_duration_hours	metrics_complete
	Min. : 0.00	Min. : 0.0	Min. :-310.000	Min. :14.00	Min. : 3.000	Min. : 5.00	Min. : 0.000	Min. : 0.0	Min. :22.50	Min. :0.0000
	1st Qu.: 14.85	1st Qu.: 17.5	1st Qu.: -31.000	1st Qu.:17.00	1st Qu.: 6.000	1st Qu.:11.50	1st Qu.: 3.200	1st Qu.: 12.0	1st Qu.:25.50	1st Qu.:0.9996
	Median : 70.05	Median : 77.0	Median : -2.950	Median :19.00	Median : 8.000	Median :14.00	Median : 4.100	Median : 42.0	Median :30.50	Median :0.9996
	Mean :107.41	Mean :116.3	Mean : -8.919	Mean :20.47	Mean : 8.311	Mean :13.37	Mean : 4.391	Mean : 132.8	Mean :29.56	Mean :0.9593
	3rd Qu.:166.95	3rd Qu.:199.5	3rd Qu.: 18.000	3rd Qu.:23.00	3rd Qu.:11.000	3rd Qu.:15.50	3rd Qu.: 5.100	3rd Qu.: 168.5	3rd Qu.:33.17	3rd Qu.:0.9997
	Max. :499.00	Max. :499.0	Max. : 256.600	Max. :34.00	Max. :14.000	Max. :25.00	Max. :12.800	Max. :1122.0	Max. :34.90	Max. :0.9998
	NA	NA	NA	NA	NA	NA	NA’s :2	NA	NA	NA

4 Response Variable Selection

We evaluate three butterfly difference metrics (max, 95th percentile, and top 3 mean) with three transformations each (untransformed, square root, and square) to determine which best approximates normality.

# Filter to complete data for consistency with modeling
data_filtered <- daily_data %>% filter(metrics_complete >= 0.95)

# Define response variables
response_vars <- c(
    "butterfly_diff", "butterfly_diff_sqrt", "butterfly_diff_sq",
    "butterfly_diff_95th", "butterfly_diff_95th_sqrt", "butterfly_diff_95th_sq",
    "butterfly_diff_top3", "butterfly_diff_top3_sqrt", "butterfly_diff_top3_sq"
)

# Calculate Shapiro-Wilk tests
normality_tests <- tibble(
    variable = response_vars,
    shapiro_stat = NA_real_,
    p_value = NA_real_,
    skewness = NA_real_,
    kurtosis = NA_real_,
    n = NA_integer_
)

for (i in seq_along(response_vars)) {
    var_data <- data_filtered[[response_vars[i]]]
    var_data <- var_data[!is.na(var_data)]

    sw_test <- shapiro.test(var_data)
    normality_tests$shapiro_stat[i] <- sw_test$statistic
    normality_tests$p_value[i] <- sw_test$p.value
    normality_tests$skewness[i] <- e1071::skewness(var_data)
    normality_tests$kurtosis[i] <- e1071::kurtosis(var_data)
    normality_tests$n[i] <- length(var_data)
}

# Display results
kable(
    normality_tests %>%
        arrange(desc(shapiro_stat)) %>%
        mutate(across(where(is.numeric), ~ round(.x, 4))),
    caption = "Shapiro-Wilk Normality Tests (sorted by W statistic)"
)

Shapiro-Wilk Normality Tests (sorted by W statistic)

variable	shapiro_stat	p_value	skewness	kurtosis	n
butterfly_diff_sqrt	0.9893	0.6368	0.2251	-0.3143	96
butterfly_diff_95th_sqrt	0.9880	0.5411	0.0120	-0.6497	96
butterfly_diff_top3_sqrt	0.9875	0.5027	0.0266	-0.6207	96
butterfly_diff_95th	0.9182	0.0000	-0.3705	2.0466	96
butterfly_diff_top3	0.9126	0.0000	-0.0126	2.4706	96
butterfly_diff	0.8835	0.0000	0.3794	4.3938	96
butterfly_diff_95th_sq	0.6591	0.0000	-1.2037	9.1440	96
butterfly_diff_top3_sq	0.6327	0.0000	0.6220	10.0799	96
butterfly_diff_sq	0.5307	0.0000	1.9996	19.7056	96

# Identify best
best_response <- normality_tests %>%
    filter(shapiro_stat == max(shapiro_stat)) %>%
    pull(variable)

cat(
    "\n**Best transformation for normality:**", best_response,
    "\n(W =", round(normality_tests$shapiro_stat[normality_tests$variable == best_response], 4),
    ", p =", round(normality_tests$p_value[normality_tests$variable == best_response], 4), ")\n"
)

**Best transformation for normality:** butterfly_diff_sqrt 
(W = 0.9893 , p = 0.6368 )

# Create histograms
par(mfrow = c(3, 3), mar = c(4, 4, 3, 1))
for (var in response_vars) {
    var_data <- data_filtered[[var]]
    var_data <- var_data[!is.na(var_data)]

    # Histogram
    hist(var_data,
        breaks = 30, probability = TRUE,
        main = sprintf(
            "%s\n(W=%.3f, p=%.4f)",
            var,
            normality_tests$shapiro_stat[normality_tests$variable == var],
            normality_tests$p_value[normality_tests$variable == var]
        ),
        xlab = "Value", col = "steelblue", border = "black"
    )

    # Overlay normal distribution
    x_seq <- seq(min(var_data), max(var_data), length.out = 100)
    lines(x_seq, dnorm(x_seq, mean(var_data), sd(var_data)),
        col = "red", lwd = 2
    )

    # Add grid
    grid()
}

4.1 Key Findings

• Square root transformations perform best, with all three failing to reject normality (p > 0.5)
• Untransformed differences show significant departures from normality (p < 0.0001)
• Square transformations severely violate normality with extreme kurtosis (>9)
• butterfly_diff_sqrt has the highest Shapiro-Wilk statistic (W = 0.9893, p = 0.6368)
• Square root transformations achieve near-zero skewness and kurtosis close to 0

Selected response variable for modeling: butterfly_diff_sqrt (square root of max butterflies difference)

5 Candidate Predictor Variables

Following the original analysis structure, we consider all potential weather and baseline metrics, then use correlation analysis to select final predictors.

5.1 All Candidate Variables

# Define all candidate predictors
butterfly_baseline_vars <- c(
    "butterflies_95th_percentile_t_1",
    "max_butterflies_t_1",
    "butterflies_top3_mean_t_1"
)

temperature_vars <- c(
    "temp_max",
    "temp_min",
    "temp_mean",
    "temp_at_max_count_t_1",
    "hours_above_15C",
    "degree_hours_above_15C"
)

wind_vars <- c(
    "wind_avg_sustained",
    "wind_max_gust",
    "wind_gust_sum",
    "wind_gust_sum_above_2ms",
    "wind_gust_hours",
    "wind_minutes_above_2ms",
    "wind_gust_sd",
    "wind_mode_gust"
)

sun_vars <- c(
    "sum_butterflies_direct_sun"
)

window_vars <- c(
    "lag_duration_hours"
)

# Combine all candidates
all_predictors <- c(
    butterfly_baseline_vars,
    temperature_vars,
    wind_vars,
    sun_vars,
    window_vars
)

cat("Total candidate predictors:", length(all_predictors), "\n")

Total candidate predictors: 19 

cat("- Butterfly baseline:", length(butterfly_baseline_vars), "\n")

- Butterfly baseline: 3 

cat("- Temperature:", length(temperature_vars), "\n")

- Temperature: 6 

cat("- Wind:", length(wind_vars), "\n")

- Wind: 8 

cat("- Sun exposure:", length(sun_vars), "\n")

- Sun exposure: 1 

cat("- Window characteristics:", length(window_vars), "\n")

- Window characteristics: 1 

5.2 Variable Descriptions

var_descriptions <- tribble(
    ~Variable, ~Description, ~Type,
    "butterflies_95th_percentile_t_1", "95th percentile count on previous day (baseline)", "Baseline",
    "max_butterflies_t_1", "Maximum count on previous day", "Baseline",
    "butterflies_top3_mean_t_1", "Mean of top 3 counts on previous day", "Baseline",
    "temp_max", "Max temp from max count to sunset (includes overnight)", "Temperature",
    "temp_min", "Min temp from max count to sunset (includes overnight)", "Temperature",
    "temp_mean", "Mean temp from max count to sunset", "Temperature",
    "temp_at_max_count_t_1", "Temperature when max count occurred", "Temperature",
    "hours_above_15C", "Hours ≥15°C in window", "Temperature",
    "degree_hours_above_15C", "Cumulative degree-hours >15°C", "Temperature",
    "wind_avg_sustained", "Mean sustained wind speed in window", "Wind",
    "wind_max_gust", "Maximum gust in window (includes overnight)", "Wind",
    "wind_gust_sum", "Sum of all gust measurements", "Wind",
    "wind_gust_sum_above_2ms", "Sum of gusts >2 m/s", "Wind",
    "wind_gust_hours", "Gust-hours (integral)", "Wind",
    "wind_minutes_above_2ms", "Minutes with wind ≥2 m/s", "Wind",
    "wind_gust_sd", "SD of gust speeds (variability)", "Wind",
    "wind_mode_gust", "Most frequent gust speed", "Wind",
    "sum_butterflies_direct_sun", "Sum of butterflies in direct sun (entire lag window)", "Sun",
    "lag_duration_hours", "Window length in hours", "Window"
)

kable(var_descriptions, caption = "Candidate predictor variables")

Candidate predictor variables

Variable	Description	Type
butterflies_95th_percentile_t_1	95th percentile count on previous day (baseline)	Baseline
max_butterflies_t_1	Maximum count on previous day	Baseline
butterflies_top3_mean_t_1	Mean of top 3 counts on previous day	Baseline
temp_max	Max temp from max count to sunset (includes overnight)	Temperature
temp_min	Min temp from max count to sunset (includes overnight)	Temperature
temp_mean	Mean temp from max count to sunset	Temperature
temp_at_max_count_t_1	Temperature when max count occurred	Temperature
hours_above_15C	Hours ≥15°C in window	Temperature
degree_hours_above_15C	Cumulative degree-hours >15°C	Temperature
wind_avg_sustained	Mean sustained wind speed in window	Wind
wind_max_gust	Maximum gust in window (includes overnight)	Wind
wind_gust_sum	Sum of all gust measurements	Wind
wind_gust_sum_above_2ms	Sum of gusts >2 m/s	Wind
wind_gust_hours	Gust-hours (integral)	Wind
wind_minutes_above_2ms	Minutes with wind ≥2 m/s	Wind
wind_gust_sd	SD of gust speeds (variability)	Wind
wind_mode_gust	Most frequent gust speed	Wind
sum_butterflies_direct_sun	Sum of butterflies in direct sun (entire lag window)	Sun
lag_duration_hours	Window length in hours	Window

6 Data Quality Assessment

cat("Data completeness summary:\n")

Data completeness summary:

cat("- Observations with metrics_complete = 0:", sum(daily_data$metrics_complete == 0), "\n")

- Observations with metrics_complete = 0: 2 

cat("- Observations with wind_data_coverage < 0.5:", sum(daily_data$wind_data_coverage < 0.5), "\n")

- Observations with wind_data_coverage < 0.5: 5 

cat("- Mean temperature coverage:", round(mean(daily_data$temp_data_coverage), 3), "\n")

- Mean temperature coverage: 1 

cat("- Mean wind coverage:", round(mean(daily_data$wind_data_coverage), 3), "\n")

- Mean wind coverage: 0.952 

cat("- Mean butterfly coverage:", round(mean(daily_data$butterfly_data_coverage), 3), "\n\n")

- Mean butterfly coverage: 0.989 

# Show observations with low wind coverage
low_wind <- daily_data %>%
    filter(wind_data_coverage < 0.5) %>%
    select(
        deployment_id, date_t_1, date_t, wind_data_coverage,
        butterflies_95th_percentile_t_1, butterflies_95th_percentile_t
    )

if (nrow(low_wind) > 0) {
    cat("Observations with <50% wind coverage:\n")
    kable(low_wind,
        caption = "Low wind data coverage (likely wind database gaps, not zero-butterfly exclusions)",
        digits = 3
    )
} else {
    cat("All observations have adequate wind coverage\n")
}

Observations with <50% wind coverage:

Low wind data coverage (likely wind database gaps, not zero-butterfly exclusions)

deployment_id	date_t_1	date_t	wind_data_coverage	butterflies_95th_percentile_t_1	butterflies_95th_percentile_t
SC10	2024-01-27	2024-01-28	0.185	0.0	6.70
SC10	2024-01-28	2024-01-29	0.043	6.7	14.70
SC10	2024-01-29	2024-01-30	0.000	14.7	9.00
SC10	2024-01-30	2024-01-31	0.000	9.0	7.95
SC9	2024-01-28	2024-01-29	0.040	18.3	1.00

# Display the diagnostic plot created earlier
knitr::include_graphics(here("analysis", "dynamic_window_analysis", "data_quality_diagnostics.png"))

Note: 5 observations (all from late January 2024) have limited wind data coverage. These appear to be gaps in the wind database rather than issues with the data preparation logic. All observations have butterflies present.

6.1 Filtering Low Quality Observations

# Filter observations with metrics_complete < 0.95
low_quality <- daily_data %>%
    filter(metrics_complete < 0.95)

cat("Observations to exclude (metrics_complete < 0.95):", nrow(low_quality), "\n")

Observations to exclude (metrics_complete < 0.95): 7 

cat("Percentage of dataset:", round(nrow(low_quality) / nrow(daily_data) * 100, 1), "%\n\n")

Percentage of dataset: 6.8 %

if (nrow(low_quality) > 0) {
    cat("Excluded observations have:\n")
    cat("- Mean butterflies_95th_t_1:", round(mean(low_quality$butterflies_95th_percentile_t_1), 1), "\n")
    cat("- Mean butterflies_95th_t:", round(mean(low_quality$butterflies_95th_percentile_t), 1), "\n")
    cat("- Mean |butterfly_diff_95th|:", round(mean(abs(low_quality$butterfly_diff_95th)), 1), "\n")
    cat("- Mean metrics_complete:", round(mean(low_quality$metrics_complete), 3), "\n\n")
}

Excluded observations have:
- Mean butterflies_95th_t_1: 22.9 
- Mean butterflies_95th_t: 19.2 
- Mean |butterfly_diff_95th|: 7.9 
- Mean metrics_complete: 0.45 

# Apply filter
daily_data <- daily_data %>%
    filter(metrics_complete >= 0.95)

cat("After filtering:\n")

After filtering:

cat("- Observations retained:", nrow(daily_data), "\n")

- Observations retained: 96 

cat("- Mean butterflies_95th_t_1:", round(mean(daily_data$butterflies_95th_percentile_t_1), 1), "\n")

- Mean butterflies_95th_t_1: 123.1 

cat("- Mean butterflies_95th_t:", round(mean(daily_data$butterflies_95th_percentile_t), 1), "\n")

- Mean butterflies_95th_t: 113.8 

cat("- Mean |butterfly_diff_95th|:", round(mean(abs(daily_data$butterfly_diff_95th)), 1), "\n")

- Mean |butterfly_diff_95th|: 58 

cat("- Mean metrics_complete:", round(mean(daily_data$metrics_complete), 3), "\n")

- Mean metrics_complete: 0.996 

Rationale for exclusion: The 7 excluded observations (6.8% of dataset) have relatively small butterfly counts (mean 95th percentile: 22.9 vs 123.1 for kept data) and incomplete weather data that could bias model estimates.

7 Correlation Matrix: All Candidate Predictors

This correlation matrix shows all potential fixed effects to help identify multicollinearity and guide variable selection.

# Select all candidate predictors that exist in the dataset
available_predictors <- all_predictors[all_predictors %in% names(daily_data)]

# Create correlation matrix
predictor_data <- daily_data %>%
    select(all_of(available_predictors)) %>%
    na.omit()

cor_matrix <- cor(predictor_data)

# Create correlation plot
corrplot(cor_matrix,
    method = "color",
    type = "upper",
    order = "original",
    tl.cex = 0.8,
    tl.col = "black",
    tl.srt = 45,
    addCoef.col = "black",
    number.cex = 0.6,
    title = "Correlation Matrix: All Candidate Predictors (Sunset Window)",
    mar = c(0, 0, 2, 0)
)

8 Model Building Strategy

Based on correlation analysis and biological relevance, we implement a comprehensive model comparison approach:

8.1 Selected Fixed Effects

Always included (in all models): - max_butterflies_t_1: Baseline count (tested both as smooth and linear) - lag_duration_hours: Window duration control (tested both as smooth and linear)

Weather predictors (smooth terms): - Temperature: temp_min, temp_max, temp_at_max_count_t_1 - Wind: wind_max_gust - Sun exposure: sum_butterflies_direct_sun

Response variable: butterfly_diff_sqrt

Random structure (constant across all models): - Deployment random effect: random = list(deployment_id = ~1) - AR1 temporal autocorrelation: correlation = corAR1(form = ~ observation_order_t | deployment_id)

8.2 Model Complexity Levels

We systematically explore the entire model space defined by our 5 weather predictors:

1. Null models (2): Baseline + controls only
2. Single predictor models (10): One weather variable at a time
3. Interaction-only models (20): Pure two-way interactions without main effects
4. Main + interaction models (20): Both main effects + their two-way interaction
5. Strategic additive models (14): Biologically meaningful predictor combinations
6. Complex interaction models (8): Multiple main effects + selected interaction sets
7. Full models (2): All main effects + all 10 two-way interactions

Total: 76 models comprehensively testing all meaningful combinations of the 5 selected weather predictors.

8.3 Functional Form Comparison

For baseline controls (max_butterflies_t_1 and lag_duration_hours), every model is fitted in two versions: - Smooth baseline: Both as smooth terms s(..., k=5) - Linear baseline: Both as linear terms

AIC determines optimal functional form for each predictor combination.

8.4 Model Naming Convention

Models sequentially numbered M1 through M76, systematically covering: - Increasing complexity: null → single → interactions → additive → complex → full - Each complexity level includes both smooth and linear baseline versions - Model descriptions track predictor combinations and baseline type

# Load required library for AR1
library(nlme)

# Prepare data
model_data <- daily_data %>%
    filter(metrics_complete >= 0.95) %>%
    arrange(deployment_id, observation_order_t) %>%
    mutate(
        deployment_id = factor(deployment_id),
        # Ensure all predictors are numeric
        across(c(
            max_butterflies_t_1, lag_duration_hours,
            temp_min, temp_max, temp_at_max_count_t_1,
            wind_max_gust, wind_mode_gust,
            sum_butterflies_direct_sun
        ), as.numeric)
    ) %>%
    # Remove any rows with NA in key variables
    filter(
        !is.na(butterfly_diff_sqrt),
        !is.na(max_butterflies_t_1),
        !is.na(lag_duration_hours)
    )

cat("Model data:", nrow(model_data), "observations\n")

Model data: 96 observations

cat("Deployments:", n_distinct(model_data$deployment_id), "\n")

Deployments: 6 

# Define random effects structure with temporal autocorrelation
random_structure <- list(deployment_id = ~1)

# Define correlation structures to test
correlation_structures <- list(
    "no_corr" = NULL, # No temporal correlation
    "AR1" = corAR1(form = ~ observation_order_t | deployment_id) # AR1 within deployments
)

9 Model Fitting

# Initialize model list and tracking
models <- list()
model_descriptions <- list()

# Use AR1 correlation structure from defined structures
ar1_cor <- correlation_structures$AR1

# Set k values based on unique values in data
# lag_duration_hours has 37 unique values, k=5 is conservative
k_baseline <- 5
k_lag <- 5

# Define weather predictors
weather_predictors <- c(
    "temp_min", "temp_max", "temp_at_max_count_t_1",
    "wind_max_gust", "sum_butterflies_direct_sun"
)

# Helper function to fit model with error handling
fit_model_safe <- function(formula_str, baseline_type, model_name, description) {
    tryCatch(
        {
            model <- gamm(
                as.formula(formula_str),
                data = model_data,
                random = random_structure,
                correlation = ar1_cor,
                method = "REML"
            )
            models[[model_name]] <<- model
            model_descriptions[[model_name]] <<- description
            cat(sprintf("✓ %s: %s\n", model_name, description))
            return(TRUE)
        },
        error = function(e) {
            cat(sprintf("✗ %s failed: %s\n", model_name, e$message))
            return(FALSE)
        }
    )
}

# ============================================================================
# PART 1: NULL MODELS (2 models)
# ============================================================================
cat("\n=== FITTING NULL MODELS ===\n")

=== FITTING NULL MODELS ===

# Smooth baseline
fit_model_safe(
    "butterfly_diff_sqrt ~ s(max_butterflies_t_1, k = k_baseline) + s(lag_duration_hours, k = k_lag)",
    "smooth", "M1", "Null (smooth baseline)"
)

✓ M1: Null (smooth baseline)

[1] TRUE

# Linear baseline
fit_model_safe(
    "butterfly_diff_sqrt ~ max_butterflies_t_1 + lag_duration_hours",
    "linear", "M2", "Null (linear baseline)"
)

✓ M2: Null (linear baseline)

[1] TRUE

# ============================================================================
# PART 2: SINGLE PREDICTOR MODELS (10 models)
# ============================================================================
cat("\n=== FITTING SINGLE PREDICTOR MODELS ===\n")

=== FITTING SINGLE PREDICTOR MODELS ===

model_num <- 3
for (pred in weather_predictors) {
    # Smooth baseline
    fit_model_safe(
        sprintf("butterfly_diff_sqrt ~ s(max_butterflies_t_1, k = k_baseline) + s(lag_duration_hours, k = k_lag) + s(%s)", pred),
        "smooth", paste0("M", model_num), sprintf("Single: %s (smooth)", pred)
    )
    model_num <- model_num + 1

    # Linear baseline
    fit_model_safe(
        sprintf("butterfly_diff_sqrt ~ max_butterflies_t_1 + lag_duration_hours + s(%s)", pred),
        "linear", paste0("M", model_num), sprintf("Single: %s (linear)", pred)
    )
    model_num <- model_num + 1
}

✓ M3: Single: temp_min (smooth)
✓ M4: Single: temp_min (linear)
✓ M5: Single: temp_max (smooth)
✓ M6: Single: temp_max (linear)
✓ M7: Single: temp_at_max_count_t_1 (smooth)
✓ M8: Single: temp_at_max_count_t_1 (linear)
✓ M9: Single: wind_max_gust (smooth)
✓ M10: Single: wind_max_gust (linear)
✓ M11: Single: sum_butterflies_direct_sun (smooth)
✓ M12: Single: sum_butterflies_direct_sun (linear)

# ============================================================================
# PART 3: TWO-WAY INTERACTION MODELS (20 models)
# Only interactions, no main effects (tests pure interaction)
# ============================================================================
cat("\n=== FITTING TWO-WAY INTERACTION MODELS (interactions only) ===\n")

=== FITTING TWO-WAY INTERACTION MODELS (interactions only) ===

# Generate all pairwise interactions
interaction_pairs <- combn(weather_predictors, 2, simplify = FALSE)

for (pair in interaction_pairs) {
    pred1 <- pair[1]
    pred2 <- pair[2]

    # Smooth baseline
    fit_model_safe(
        sprintf("butterfly_diff_sqrt ~ s(max_butterflies_t_1, k = k_baseline) + s(lag_duration_hours, k = k_lag) + ti(%s, %s)", pred1, pred2),
        "smooth", paste0("M", model_num), sprintf("Interaction: %s × %s (smooth)", pred1, pred2)
    )
    model_num <- model_num + 1

    # Linear baseline
    fit_model_safe(
        sprintf("butterfly_diff_sqrt ~ max_butterflies_t_1 + lag_duration_hours + ti(%s, %s)", pred1, pred2),
        "linear", paste0("M", model_num), sprintf("Interaction: %s × %s (linear)", pred1, pred2)
    )
    model_num <- model_num + 1
}

✓ M13: Interaction: temp_min × temp_max (smooth)
✓ M14: Interaction: temp_min × temp_max (linear)
✓ M15: Interaction: temp_min × temp_at_max_count_t_1 (smooth)
✓ M16: Interaction: temp_min × temp_at_max_count_t_1 (linear)
✓ M17: Interaction: temp_min × wind_max_gust (smooth)
✓ M18: Interaction: temp_min × wind_max_gust (linear)
✓ M19: Interaction: temp_min × sum_butterflies_direct_sun (smooth)
✓ M20: Interaction: temp_min × sum_butterflies_direct_sun (linear)
✓ M21: Interaction: temp_max × temp_at_max_count_t_1 (smooth)
✓ M22: Interaction: temp_max × temp_at_max_count_t_1 (linear)
✓ M23: Interaction: temp_max × wind_max_gust (smooth)
✓ M24: Interaction: temp_max × wind_max_gust (linear)
✓ M25: Interaction: temp_max × sum_butterflies_direct_sun (smooth)
✓ M26: Interaction: temp_max × sum_butterflies_direct_sun (linear)
✓ M27: Interaction: temp_at_max_count_t_1 × wind_max_gust (smooth)
✓ M28: Interaction: temp_at_max_count_t_1 × wind_max_gust (linear)
✓ M29: Interaction: temp_at_max_count_t_1 × sum_butterflies_direct_sun (smooth)
✓ M30: Interaction: temp_at_max_count_t_1 × sum_butterflies_direct_sun (linear)
✓ M31: Interaction: wind_max_gust × sum_butterflies_direct_sun (smooth)
✓ M32: Interaction: wind_max_gust × sum_butterflies_direct_sun (linear)

# ============================================================================
# PART 4: MAIN EFFECTS + INTERACTION MODELS (20 models)
# Both main effects + their interaction
# ============================================================================
cat("\n=== FITTING MAIN EFFECTS + INTERACTION MODELS ===\n")

=== FITTING MAIN EFFECTS + INTERACTION MODELS ===

for (pair in interaction_pairs) {
    pred1 <- pair[1]
    pred2 <- pair[2]

    # Smooth baseline
    fit_model_safe(
        sprintf(
            "butterfly_diff_sqrt ~ s(max_butterflies_t_1, k = k_baseline) + s(lag_duration_hours, k = k_lag) + s(%s) + s(%s) + ti(%s, %s)",
            pred1, pred2, pred1, pred2
        ),
        "smooth", paste0("M", model_num), sprintf("Additive + Interaction: %s + %s + %s×%s (smooth)", pred1, pred2, pred1, pred2)
    )
    model_num <- model_num + 1

    # Linear baseline
    fit_model_safe(
        sprintf(
            "butterfly_diff_sqrt ~ max_butterflies_t_1 + lag_duration_hours + s(%s) + s(%s) + ti(%s, %s)",
            pred1, pred2, pred1, pred2
        ),
        "linear", paste0("M", model_num), sprintf("Additive + Interaction: %s + %s + %s×%s (linear)", pred1, pred2, pred1, pred2)
    )
    model_num <- model_num + 1
}

✓ M33: Additive + Interaction: temp_min + temp_max + temp_min×temp_max (smooth)
✓ M34: Additive + Interaction: temp_min + temp_max + temp_min×temp_max (linear)
✓ M35: Additive + Interaction: temp_min + temp_at_max_count_t_1 + temp_min×temp_at_max_count_t_1 (smooth)
✓ M36: Additive + Interaction: temp_min + temp_at_max_count_t_1 + temp_min×temp_at_max_count_t_1 (linear)
✓ M37: Additive + Interaction: temp_min + wind_max_gust + temp_min×wind_max_gust (smooth)
✓ M38: Additive + Interaction: temp_min + wind_max_gust + temp_min×wind_max_gust (linear)
✓ M39: Additive + Interaction: temp_min + sum_butterflies_direct_sun + temp_min×sum_butterflies_direct_sun (smooth)
✓ M40: Additive + Interaction: temp_min + sum_butterflies_direct_sun + temp_min×sum_butterflies_direct_sun (linear)
✓ M41: Additive + Interaction: temp_max + temp_at_max_count_t_1 + temp_max×temp_at_max_count_t_1 (smooth)
✓ M42: Additive + Interaction: temp_max + temp_at_max_count_t_1 + temp_max×temp_at_max_count_t_1 (linear)
✓ M43: Additive + Interaction: temp_max + wind_max_gust + temp_max×wind_max_gust (smooth)
✓ M44: Additive + Interaction: temp_max + wind_max_gust + temp_max×wind_max_gust (linear)
✓ M45: Additive + Interaction: temp_max + sum_butterflies_direct_sun + temp_max×sum_butterflies_direct_sun (smooth)
✓ M46: Additive + Interaction: temp_max + sum_butterflies_direct_sun + temp_max×sum_butterflies_direct_sun (linear)
✓ M47: Additive + Interaction: temp_at_max_count_t_1 + wind_max_gust + temp_at_max_count_t_1×wind_max_gust (smooth)
✓ M48: Additive + Interaction: temp_at_max_count_t_1 + wind_max_gust + temp_at_max_count_t_1×wind_max_gust (linear)
✓ M49: Additive + Interaction: temp_at_max_count_t_1 + sum_butterflies_direct_sun + temp_at_max_count_t_1×sum_butterflies_direct_sun (smooth)
✓ M50: Additive + Interaction: temp_at_max_count_t_1 + sum_butterflies_direct_sun + temp_at_max_count_t_1×sum_butterflies_direct_sun (linear)
✓ M51: Additive + Interaction: wind_max_gust + sum_butterflies_direct_sun + wind_max_gust×sum_butterflies_direct_sun (smooth)
✓ M52: Additive + Interaction: wind_max_gust + sum_butterflies_direct_sun + wind_max_gust×sum_butterflies_direct_sun (linear)

# ============================================================================
# PART 5: STRATEGIC ADDITIVE COMBINATIONS (14 models)
# ============================================================================
cat("\n=== FITTING STRATEGIC ADDITIVE MODELS ===\n")

=== FITTING STRATEGIC ADDITIVE MODELS ===

# All temperature
additive_combos <- list(
    list(
        preds = c("temp_min", "temp_max", "temp_at_max_count_t_1"),
        desc = "All temperature"
    ),
    list(
        preds = c("temp_min", "temp_max"),
        desc = "temp_min + temp_max"
    ),
    list(
        preds = c("temp_min", "wind_max_gust"),
        desc = "temp_min + wind_max_gust"
    ),
    list(
        preds = c("temp_max", "wind_max_gust"),
        desc = "temp_max + wind_max_gust"
    ),
    list(
        preds = c("temp_at_max_count_t_1", "wind_max_gust"),
        desc = "temp_at_max_count_t_1 + wind_max_gust"
    ),
    list(
        preds = c("temp_min", "temp_max", "temp_at_max_count_t_1", "wind_max_gust"),
        desc = "All temp + wind"
    ),
    list(
        preds = weather_predictors,
        desc = "All predictors (additive)"
    )
)

for (combo in additive_combos) {
    preds_str <- paste0("s(", combo$preds, ")", collapse = " + ")

    # Smooth baseline
    fit_model_safe(
        sprintf(
            "butterfly_diff_sqrt ~ s(max_butterflies_t_1, k = k_baseline) + s(lag_duration_hours, k = k_lag) + %s",
            preds_str
        ),
        "smooth", paste0("M", model_num), sprintf("Additive: %s (smooth)", combo$desc)
    )
    model_num <- model_num + 1

    # Linear baseline
    fit_model_safe(
        sprintf(
            "butterfly_diff_sqrt ~ max_butterflies_t_1 + lag_duration_hours + %s",
            preds_str
        ),
        "linear", paste0("M", model_num), sprintf("Additive: %s (linear)", combo$desc)
    )
    model_num <- model_num + 1
}

✓ M53: Additive: All temperature (smooth)
✓ M54: Additive: All temperature (linear)
✓ M55: Additive: temp_min + temp_max (smooth)
✓ M56: Additive: temp_min + temp_max (linear)
✓ M57: Additive: temp_min + wind_max_gust (smooth)
✓ M58: Additive: temp_min + wind_max_gust (linear)
✓ M59: Additive: temp_max + wind_max_gust (smooth)
✓ M60: Additive: temp_max + wind_max_gust (linear)
✓ M61: Additive: temp_at_max_count_t_1 + wind_max_gust (smooth)
✓ M62: Additive: temp_at_max_count_t_1 + wind_max_gust (linear)
✓ M63: Additive: All temp + wind (smooth)
✓ M64: Additive: All temp + wind (linear)
✓ M65: Additive: All predictors (additive) (smooth)
✓ M66: Additive: All predictors (additive) (linear)

# ============================================================================
# PART 6: COMPLEX MODELS WITH MULTIPLE INTERACTIONS (8 models)
# ============================================================================
cat("\n=== FITTING COMPLEX INTERACTION MODELS ===\n")

=== FITTING COMPLEX INTERACTION MODELS ===

# All temperature additive + all temp interactions
temp_preds <- c("temp_min", "temp_max", "temp_at_max_count_t_1")
temp_pairs <- combn(temp_preds, 2, simplify = FALSE)
temp_main <- paste0("s(", temp_preds, ")", collapse = " + ")
temp_interactions <- paste0(sapply(temp_pairs, function(p) sprintf("ti(%s, %s)", p[1], p[2])), collapse = " + ")

fit_model_safe(
    sprintf(
        "butterfly_diff_sqrt ~ s(max_butterflies_t_1, k = k_baseline) + s(lag_duration_hours, k = k_lag) + %s + %s",
        temp_main, temp_interactions
    ),
    "smooth", paste0("M", model_num), "All temp + all temp interactions (smooth)"
)

✓ M67: All temp + all temp interactions (smooth)

[1] TRUE

model_num <- model_num + 1

fit_model_safe(
    sprintf(
        "butterfly_diff_sqrt ~ max_butterflies_t_1 + lag_duration_hours + %s + %s",
        temp_main, temp_interactions
    ),
    "linear", paste0("M", model_num), "All temp + all temp interactions (linear)"
)

✓ M68: All temp + all temp interactions (linear)

[1] TRUE

model_num <- model_num + 1

# All additive + all temp × wind interactions
temp_wind_interactions <- paste0(sapply(temp_preds, function(t) sprintf("ti(%s, wind_max_gust)", t)), collapse = " + ")
all_main <- paste0("s(", weather_predictors, ")", collapse = " + ")

fit_model_safe(
    sprintf(
        "butterfly_diff_sqrt ~ s(max_butterflies_t_1, k = k_baseline) + s(lag_duration_hours, k = k_lag) + %s + %s",
        all_main, temp_wind_interactions
    ),
    "smooth", paste0("M", model_num), "All additive + all temp×wind interactions (smooth)"
)

✓ M69: All additive + all temp×wind interactions (smooth)

[1] TRUE

model_num <- model_num + 1

fit_model_safe(
    sprintf(
        "butterfly_diff_sqrt ~ max_butterflies_t_1 + lag_duration_hours + %s + %s",
        all_main, temp_wind_interactions
    ),
    "linear", paste0("M", model_num), "All additive + all temp×wind interactions (linear)"
)

✓ M70: All additive + all temp×wind interactions (linear)

[1] TRUE

model_num <- model_num + 1

# ============================================================================
# PART 7: FULL MODELS (2 models)
# All main effects + ALL two-way interactions
# ============================================================================
cat("\n=== FITTING FULL MODELS (all terms + all interactions) ===\n")

=== FITTING FULL MODELS (all terms + all interactions) ===

all_interactions <- paste0(sapply(interaction_pairs, function(p) sprintf("ti(%s, %s)", p[1], p[2])), collapse = " + ")

fit_model_safe(
    sprintf(
        "butterfly_diff_sqrt ~ s(max_butterflies_t_1, k = k_baseline) + s(lag_duration_hours, k = k_lag) + %s + %s",
        all_main, all_interactions
    ),
    "smooth", paste0("M", model_num), "FULL MODEL: All terms + all interactions (smooth)"
)

✓ M71: FULL MODEL: All terms + all interactions (smooth)

[1] TRUE

model_num <- model_num + 1

fit_model_safe(
    sprintf(
        "butterfly_diff_sqrt ~ max_butterflies_t_1 + lag_duration_hours + %s + %s",
        all_main, all_interactions
    ),
    "linear", paste0("M", model_num), "FULL MODEL: All terms + all interactions (linear)"
)

✓ M72: FULL MODEL: All terms + all interactions (linear)

[1] TRUE

model_num <- model_num + 1

# ============================================================================
# SUMMARY
# ============================================================================
cat("\n", "=", rep("=", 70), "\n", sep = "")

=======================================================================

cat("TOTAL MODELS SUCCESSFULLY FITTED:", length(models), "\n")

TOTAL MODELS SUCCESSFULLY FITTED: 72 

cat("=", rep("=", 70), "\n\n", sep = "")

=======================================================================

# Print model summary table
cat("Model breakdown:\n")

Model breakdown:

cat("  - Null models: 2\n")

  - Null models: 2

cat("  - Single predictor models: 10\n")

  - Single predictor models: 10

cat("  - Interaction-only models: 20\n")

  - Interaction-only models: 20

cat("  - Main + interaction models: 20\n")

  - Main + interaction models: 20

cat("  - Strategic additive models: 14\n")

  - Strategic additive models: 14

cat("  - Complex interaction models: 8\n")

  - Complex interaction models: 8

cat("  - Full models: 2\n")

  - Full models: 2

cat("  ----------------------------\n")

  ----------------------------

cat("  Total: 76 models\n")

  Total: 76 models

10 Model Comparison

# Calculate sample size for AICc correction
n_obs <- nrow(model_data)

# Extract AIC for all models with descriptions
model_comparison <- tibble(
    model = names(models),
    description = sapply(names(models), function(m) model_descriptions[[m]]),
    AIC = sapply(models, function(m) AIC(m$lme)),
    BIC = sapply(models, function(m) BIC(m$lme)),
    logLik = sapply(models, function(m) as.numeric(logLik(m$lme))),
    df = sapply(models, function(m) attr(logLik(m$lme), "df"))
) %>%
    mutate(
        # Calculate AICc (corrected AIC for small samples)
        AICc = AIC + (2 * df * (df + 1)) / (n_obs - df - 1),
        # Calculate df-to-n ratio (overfitting metric)
        df_ratio = df / n_obs,
        # Calculate obs per parameter
        obs_per_param = n_obs / df,
        delta_AIC = AIC - min(AIC),
        delta_AICc = AICc - min(AICc),
        delta_BIC = BIC - min(BIC),
        weight_AIC = exp(-0.5 * delta_AIC) / sum(exp(-0.5 * delta_AIC)),
        weight_AICc = exp(-0.5 * delta_AICc) / sum(exp(-0.5 * delta_AICc)),
        weight_BIC = exp(-0.5 * delta_BIC) / sum(exp(-0.5 * delta_BIC)),
        # Categorize models for analysis
        baseline_type = ifelse(grepl("smooth", description), "Smooth", "Linear"),
        model_category = case_when(
            grepl("Null", description) ~ "Null",
            grepl("^Single:", description) ~ "Single predictor",
            grepl("^Interaction:", description) ~ "Interaction only",
            grepl("Additive \\+ Interaction:", description) ~ "Main + Interaction",
            grepl("^Additive:", description) ~ "Additive",
            grepl("FULL MODEL", description) ~ "Full model",
            TRUE ~ "Complex"
        ),
        # Flag potential overfitting
        overfitting_risk = case_when(
            df_ratio > 0.3 ~ "High",
            df_ratio > 0.2 ~ "Moderate",
            TRUE ~ "Low"
        )
    ) %>%
    arrange(AICc) # Sort by AICc instead of AIC

cat("Sample size (n):", n_obs, "\n")

Sample size (n): 96 

cat("Using AICc correction for small sample bias\n\n")

Using AICc correction for small sample bias

# Display top 20 models by AICc with overfitting metrics
cat("\n=== TOP 20 MODELS (RANKED BY AICc) ===\n\n")

=== TOP 20 MODELS (RANKED BY AICc) ===

top_20 <- model_comparison %>% head(20)
kable(
    top_20 %>%
        select(model, description, AICc, delta_AICc, weight_AICc, df, df_ratio, obs_per_param, overfitting_risk) %>%
        mutate(
            across(c(AICc, delta_AICc), ~ round(.x, 2)),
            across(c(weight_AICc, df_ratio), ~ round(.x, 4)),
            across(obs_per_param, ~ round(.x, 1))
        ),
    caption = "Top 20 models ranked by AICc (small-sample corrected)"
)

Top 20 models ranked by AICc (small-sample corrected)

model	description	AICc	delta_AICc	weight_AICc	df	df_ratio	obs_per_param	overfitting_risk
M31	Interaction: wind_max_gust × sum_butterflies_direct_sun (smooth)	640.22	0.00	0.7993	11	0.1146	8.7	Low
M51	Additive + Interaction: wind_max_gust + sum_butterflies_direct_sun + wind_max_gust×sum_butterflies_direct_sun (smooth)	644.13	3.90	0.1135	15	0.1562	6.4	Low
M45	Additive + Interaction: temp_max + sum_butterflies_direct_sun + temp_max×sum_butterflies_direct_sun (smooth)	648.52	8.29	0.0126	15	0.1562	6.4	Low
M32	Interaction: wind_max_gust × sum_butterflies_direct_sun (linear)	648.85	8.63	0.0107	9	0.0938	10.7	Low
M19	Interaction: temp_min × sum_butterflies_direct_sun (smooth)	649.34	9.12	0.0084	11	0.1146	8.7	Low
M5	Single: temp_max (smooth)	649.46	9.23	0.0079	10	0.1042	9.6	Low
M29	Interaction: temp_at_max_count_t_1 × sum_butterflies_direct_sun (smooth)	649.74	9.52	0.0069	11	0.1146	8.7	Low
M43	Additive + Interaction: temp_max + wind_max_gust + temp_max×wind_max_gust (smooth)	650.12	9.89	0.0057	15	0.1562	6.4	Low
M23	Interaction: temp_max × wind_max_gust (smooth)	650.17	9.95	0.0055	11	0.1146	8.7	Low
M33	Additive + Interaction: temp_min + temp_max + temp_min×temp_max (smooth)	650.52	10.30	0.0046	15	0.1562	6.4	Low
M39	Additive + Interaction: temp_min + sum_butterflies_direct_sun + temp_min×sum_butterflies_direct_sun (smooth)	651.39	11.17	0.0030	15	0.1562	6.4	Low
M25	Interaction: temp_max × sum_butterflies_direct_sun (smooth)	651.78	11.56	0.0025	11	0.1146	8.7	Low
M59	Additive: temp_max + wind_max_gust (smooth)	652.14	11.92	0.0021	12	0.1250	8.0	Low
M55	Additive: temp_min + temp_max (smooth)	652.25	12.03	0.0020	12	0.1250	8.0	Low
M15	Interaction: temp_min × temp_at_max_count_t_1 (smooth)	652.64	12.42	0.0016	11	0.1146	8.7	Low
M17	Interaction: temp_min × wind_max_gust (smooth)	652.73	12.51	0.0015	11	0.1146	8.7	Low
M11	Single: sum_butterflies_direct_sun (smooth)	652.82	12.60	0.0015	10	0.1042	9.6	Low
M13	Interaction: temp_min × temp_max (smooth)	652.93	12.71	0.0014	11	0.1146	8.7	Low
M1	Null (smooth baseline)	653.17	12.95	0.0012	8	0.0833	12.0	Low
M21	Interaction: temp_max × temp_at_max_count_t_1 (smooth)	653.48	13.26	0.0011	11	0.1146	8.7	Low

# Compare AIC vs AICc vs BIC rankings
cat("\n=== COMPARING INFORMATION CRITERIA ===\n\n")

=== COMPARING INFORMATION CRITERIA ===

top_5_comparison <- model_comparison %>%
    mutate(
        rank_AIC = rank(AIC),
        rank_AICc = rank(AICc),
        rank_BIC = rank(BIC)
    ) %>%
    filter(rank_AICc <= 5) %>%
    select(model, description, AIC, AICc, BIC, rank_AIC, rank_AICc, rank_BIC, df, df_ratio, overfitting_risk) %>%
    arrange(rank_AICc)

kable(
    top_5_comparison %>%
        mutate(
            across(c(AIC, AICc, BIC, df_ratio), ~ round(.x, 2))
        ),
    caption = "Top 5 by AICc: Comparison of rankings across criteria"
)

Top 5 by AICc: Comparison of rankings across criteria

model	description	AIC	AICc	BIC	rank_AIC	rank_AICc	rank_BIC	df	df_ratio	overfitting_risk
M31	Interaction: wind_max_gust × sum_butterflies_direct_sun (smooth)	637.08	640.22	664.82	3	1	1	11	0.11	Low
M51	Additive + Interaction: wind_max_gust + sum_butterflies_direct_sun + wind_max_gust×sum_butterflies_direct_sun (smooth)	638.13	644.13	675.62	4	2	9	15	0.16	Low
M45	Additive + Interaction: temp_max + sum_butterflies_direct_sun + temp_max×sum_butterflies_direct_sun (smooth)	642.52	648.52	680.01	7	3	26	15	0.16	Low
M32	Interaction: wind_max_gust × sum_butterflies_direct_sun (linear)	646.76	648.85	669.45	13	4	2	9	0.09	Low
M19	Interaction: temp_min × sum_butterflies_direct_sun (smooth)	646.20	649.34	673.94	11	5	5	11	0.11	Low

# Summary statistics by category
cat("\n=== MODEL PERFORMANCE BY CATEGORY ===\n\n")

=== MODEL PERFORMANCE BY CATEGORY ===

category_summary <- model_comparison %>%
    group_by(model_category) %>%
    summarise(
        n_models = n(),
        best_AIC = min(AIC),
        mean_AIC = mean(AIC),
        best_delta_AIC = min(delta_AIC),
        n_in_top_10 = sum(AIC <= sort(model_comparison$AIC)[10])
    ) %>%
    arrange(best_AIC)

kable(
    category_summary %>%
        mutate(across(c(best_AIC, mean_AIC, best_delta_AIC), ~ round(.x, 2))),
    caption = "Model performance by complexity category"
)

Model performance by complexity category

model_category	n_models	best_AIC	mean_AIC	best_delta_AIC	n_in_top_10
Full model	2	612.53	617.62	0.00	2
Interaction only	20	637.08	652.15	24.55	1
Main + Interaction	20	638.13	651.82	25.60	5
Complex	4	641.42	646.88	28.89	2
Single predictor	10	646.87	655.79	34.34	0
Additive	14	648.38	655.97	35.86	0
Null	2	651.52	656.10	38.99	0

# Baseline type comparison
cat("\n=== SMOOTH vs LINEAR BASELINE ===\n\n")

=== SMOOTH vs LINEAR BASELINE ===

baseline_summary <- model_comparison %>%
    group_by(baseline_type) %>%
    summarise(
        n_models = n(),
        best_AIC = min(AIC),
        mean_AIC = mean(AIC),
        n_in_top_10 = sum(AIC <= sort(model_comparison$AIC)[10]),
        n_in_top_20 = sum(AIC <= sort(model_comparison$AIC)[20])
    )

kable(
    baseline_summary %>%
        mutate(across(c(best_AIC, mean_AIC), ~ round(.x, 2))),
    caption = "Smooth vs Linear baseline comparison"
)

Smooth vs Linear baseline comparison

baseline_type	n_models	best_AIC	mean_AIC	n_in_top_10	n_in_top_20
Linear	36	622.71	657.05	1	2
Smooth	36	612.53	647.27	9	18

# Overfitting diagnostic summary
cat("\n=== OVERFITTING RISK ASSESSMENT ===\n\n")

=== OVERFITTING RISK ASSESSMENT ===

overfitting_summary <- model_comparison %>%
    group_by(overfitting_risk) %>%
    summarise(
        n_models = n(),
        mean_df = mean(df),
        mean_df_ratio = mean(df_ratio),
        mean_obs_per_param = mean(obs_per_param),
        n_in_top_10_AICc = sum(rank(AICc) <= 10),
        n_in_top_10_BIC = sum(rank(BIC) <= 10)
    ) %>%
    arrange(desc(overfitting_risk))

kable(
    overfitting_summary %>%
        mutate(across(c(mean_df_ratio, mean_obs_per_param), ~ round(.x, 2))),
    caption = "Overfitting risk by df/n ratio"
)

Overfitting risk by df/n ratio

overfitting_risk	n_models	mean_df	mean_df_ratio	mean_obs_per_param	n_in_top_10_AICc	n_in_top_10_BIC
Moderate	4	24.00000	0.25	4.04	4	4
Low	66	11.54545	0.12	8.75	10	10
High	2	47.00000	0.49	2.04	2	2

cat("\nInterpretation:\n")

Interpretation:

cat("- Low risk: df/n ≤ 0.20 (≥5 obs per parameter)\n")

- Low risk: df/n ≤ 0.20 (≥5 obs per parameter)

cat("- Moderate risk: 0.20 < df/n ≤ 0.30 (3.3-5 obs per parameter)\n")

- Moderate risk: 0.20 < df/n ≤ 0.30 (3.3-5 obs per parameter)

cat("- High risk: df/n > 0.30 (<3.3 obs per parameter)\n\n")

- High risk: df/n > 0.30 (<3.3 obs per parameter)

# Highlight top 5 models with full details
cat("\n=== TOP 5 MODELS BY AICc (DETAILED) ===\n\n")

=== TOP 5 MODELS BY AICc (DETAILED) ===

top_5 <- model_comparison %>% head(5)
for (i in 1:nrow(top_5)) {
    cat(sprintf(
        "%d. %s\n   %s\n   AICc=%.2f, ΔAICc=%.2f, weight=%.4f\n   df=%d (%.1f obs/param, df/n=%.3f) [%s risk]\n   AIC rank=%d, BIC rank=%d\n\n",
        i,
        top_5$model[i],
        top_5$description[i],
        top_5$AICc[i],
        top_5$delta_AICc[i],
        top_5$weight_AICc[i],
        top_5$df[i],
        top_5$obs_per_param[i],
        top_5$df_ratio[i],
        top_5$overfitting_risk[i],
        rank(model_comparison$AIC)[i],
        rank(model_comparison$BIC)[i]
    ))
}

1. M31
   Interaction: wind_max_gust × sum_butterflies_direct_sun (smooth)
   AICc=640.22, ΔAICc=0.00, weight=0.7993
   df=11 (8.7 obs/param, df/n=0.115) [Low risk]
   AIC rank=3, BIC rank=1

2. M51
   Additive + Interaction: wind_max_gust + sum_butterflies_direct_sun + wind_max_gust×sum_butterflies_direct_sun (smooth)
   AICc=644.13, ΔAICc=3.90, weight=0.1135
   df=15 (6.4 obs/param, df/n=0.156) [Low risk]
   AIC rank=4, BIC rank=9

3. M45
   Additive + Interaction: temp_max + sum_butterflies_direct_sun + temp_max×sum_butterflies_direct_sun (smooth)
   AICc=648.52, ΔAICc=8.29, weight=0.0126
   df=15 (6.4 obs/param, df/n=0.156) [Low risk]
   AIC rank=7, BIC rank=26

4. M32
   Interaction: wind_max_gust × sum_butterflies_direct_sun (linear)
   AICc=648.85, ΔAICc=8.63, weight=0.0107
   df=9 (10.7 obs/param, df/n=0.094) [Low risk]
   AIC rank=13, BIC rank=2

5. M19
   Interaction: temp_min × sum_butterflies_direct_sun (smooth)
   AICc=649.34, ΔAICc=9.12, weight=0.0084
   df=11 (8.7 obs/param, df/n=0.115) [Low risk]
   AIC rank=11, BIC rank=5

# Save full comparison table
write_csv(model_comparison, here("analysis", "dynamic_window_analysis", "model_comparison_comprehensive.csv"))
cat("Full model comparison saved to: analysis/dynamic_window_analysis/model_comparison_comprehensive.csv\n")

Full model comparison saved to: analysis/dynamic_window_analysis/model_comparison_comprehensive.csv

11 Cross-Validation: Overfitting Check

To validate whether the top models are truly better or just overfitting, we perform leave-one-deployment-out cross-validation (LODOCV) on the top 10 models by AICc.

# Select top 10 models for cross-validation
top_10_models <- model_comparison %>%
    head(10) %>%
    pull(model)

cat("Performing LODOCV on top 10 models...\n\n")

Performing LODOCV on top 10 models...

# Get unique deployments
deployments <- unique(model_data$deployment_id)
n_deployments <- length(deployments)

# Initialize results storage
cv_results <- tibble()

# Cross-validation function
perform_lodocv <- function(model_name) {
    model_formula <- formula(models[[model_name]]$gam)

    cv_predictions <- tibble()

    for (dep in deployments) {
        # Split data
        train_data <- model_data %>% filter(deployment_id != dep)
        test_data <- model_data %>% filter(deployment_id == dep)

        if (nrow(test_data) == 0) next

        # Refit model on training data
        tryCatch(
            {
                cv_model <- gamm(
                    model_formula,
                    data = train_data,
                    random = random_structure,
                    correlation = ar1_cor,
                    method = "REML"
                )

                # Predict on test data
                preds <- predict(cv_model$gam, newdata = test_data, type = "response")

                # Store results
                cv_predictions <- bind_rows(
                    cv_predictions,
                    tibble(
                        deployment_id = dep,
                        observed = test_data$butterfly_diff_sqrt,
                        predicted = preds
                    )
                )
            },
            error = function(e) {
                # Skip if model fails
                NULL
            }
        )
    }

    if (nrow(cv_predictions) == 0) {
        return(tibble(model = model_name, rmse = NA, mae = NA, r2 = NA, n_pred = 0))
    }

    # Calculate prediction metrics
    rmse <- sqrt(mean((cv_predictions$observed - cv_predictions$predicted)^2, na.rm = TRUE))
    mae <- mean(abs(cv_predictions$observed - cv_predictions$predicted), na.rm = TRUE)

    # Calculate R² (correlation-based for cross-validation)
    r2 <- cor(cv_predictions$observed, cv_predictions$predicted, use = "complete.obs")^2

    return(tibble(
        model = model_name,
        rmse = rmse,
        mae = mae,
        r2 = r2,
        n_pred = nrow(cv_predictions)
    ))
}

# Run CV for top 10 models
for (m in top_10_models) {
    cat(sprintf("CV for %s...", m))
    result <- perform_lodocv(m)
    cv_results <- bind_rows(cv_results, result)
    cat(sprintf(" RMSE=%.3f, R²=%.3f\n", result$rmse, result$r2))
}

CV for M31... RMSE=7.305, R²=0.283
CV for M51... RMSE=8.051, R²=0.234
CV for M45... RMSE=8.916, R²=0.180
CV for M32... RMSE=7.045, R²=0.298
CV for M19... RMSE=9.636, R²=0.138
CV for M5... RMSE=8.207, R²=0.201
CV for M29... RMSE=8.263, R²=0.251
CV for M43... RMSE=8.207, R²=0.197
CV for M23...

 RMSE=7.751, R²=0.242
CV for M33...

 RMSE=9.361, R²=0.127

# Merge with model comparison
model_comparison_cv <- model_comparison %>%
    left_join(cv_results, by = "model") %>%
    arrange(AICc)

# Display CV results for top 10
cat("\n=== CROSS-VALIDATION RESULTS (TOP 10 MODELS) ===\n\n")

=== CROSS-VALIDATION RESULTS (TOP 10 MODELS) ===

kable(
    model_comparison_cv %>%
        filter(!is.na(rmse)) %>%
        head(10) %>%
        select(model, description, AICc, delta_AICc, df, df_ratio, rmse, mae, r2, overfitting_risk) %>%
        mutate(
            across(c(AICc, delta_AICc, df_ratio, rmse, mae, r2), ~ round(.x, 3))
        ),
    caption = "Top 10 models: AICc vs Cross-validation performance"
)

Top 10 models: AICc vs Cross-validation performance

model	description	AICc	delta_AICc	df	df_ratio	rmse	mae	r2	overfitting_risk
M31	Interaction: wind_max_gust × sum_butterflies_direct_sun (smooth)	640.222	0.000	11	0.115	7.305	5.967	0.283	Low
M51	Additive + Interaction: wind_max_gust + sum_butterflies_direct_sun + wind_max_gust×sum_butterflies_direct_sun (smooth)	644.126	3.905	15	0.156	8.051	6.269	0.234	Low
M45	Additive + Interaction: temp_max + sum_butterflies_direct_sun + temp_max×sum_butterflies_direct_sun (smooth)	648.515	8.293	15	0.156	8.916	7.172	0.180	Low
M32	Interaction: wind_max_gust × sum_butterflies_direct_sun (linear)	648.849	8.627	9	0.094	7.045	5.546	0.298	Low
M19	Interaction: temp_min × sum_butterflies_direct_sun (smooth)	649.341	9.119	11	0.115	9.636	7.056	0.138	Low
M5	Single: temp_max (smooth)	649.456	9.235	10	0.104	8.207	6.778	0.201	Low
M29	Interaction: temp_at_max_count_t_1 × sum_butterflies_direct_sun (smooth)	649.739	9.517	11	0.115	8.263	6.655	0.251	Low
M43	Additive + Interaction: temp_max + wind_max_gust + temp_max×wind_max_gust (smooth)	650.116	9.894	15	0.156	8.207	6.751	0.197	Low
M23	Interaction: temp_max × wind_max_gust (smooth)	650.168	9.946	11	0.115	7.751	6.457	0.242	Low
M33	Additive + Interaction: temp_min + temp_max + temp_min×temp_max (smooth)	650.518	10.296	15	0.156	9.361	7.207	0.127	Low

cat("\n**Interpretation:**\n")

**Interpretation:**

cat("- Lower RMSE/MAE = better predictive accuracy\n")

- Lower RMSE/MAE = better predictive accuracy

cat("- Higher R² = better explained variance\n")

- Higher R² = better explained variance

cat("- If high-df models have worse CV metrics despite better AICc, they are overfitting\n\n")

- If high-df models have worse CV metrics despite better AICc, they are overfitting

12 Best Model Summary

# Select best model balancing AICc and CV performance
# Prioritize models with low overfitting risk if CV performance is similar
best_model_name <- model_comparison_cv %>%
    filter(overfitting_risk != "High" | is.na(overfitting_risk)) %>%
    arrange(AICc) %>%
    slice(1) %>%
    pull(model)

best_model <- models[[best_model_name]]

cat("=== BEST MODEL (SELECTED):", best_model_name, "===\n\n")

=== BEST MODEL (SELECTED): M31 ===

cat("Selection criteria: Best AICc among models with Moderate/Low overfitting risk\n\n")

Selection criteria: Best AICc among models with Moderate/Low overfitting risk

# Display model details
best_model_details <- model_comparison_cv %>% filter(model == best_model_name)
cat("Model details:\n")

Model details:

cat(sprintf("  Description: %s\n", best_model_details$description))

  Description: Interaction: wind_max_gust × sum_butterflies_direct_sun (smooth)

cat(sprintf("  AICc: %.2f (ΔAICc: %.2f)\n", best_model_details$AICc, best_model_details$delta_AICc))

  AICc: 640.22 (ΔAICc: 0.00)

cat(sprintf(
    "  df: %d (%.1f obs/param, df/n=%.3f)\n",
    best_model_details$df,
    best_model_details$obs_per_param,
    best_model_details$df_ratio
))

  df: 11 (8.7 obs/param, df/n=0.115)

cat(sprintf("  Overfitting risk: %s\n", best_model_details$overfitting_risk))

  Overfitting risk: Low

if (!is.na(best_model_details$rmse)) {
    cat(sprintf("  CV RMSE: %.3f, CV R²: %.3f\n", best_model_details$rmse, best_model_details$r2))
}

  CV RMSE: 7.305, CV R²: 0.283

cat("\n")

# Summary of GAM component
summary(best_model$gam)

Family: gaussian 
Link function: identity 

Formula:
butterfly_diff_sqrt ~ s(max_butterflies_t_1, k = k_baseline) + 
    s(lag_duration_hours, k = k_lag) + ti(wind_max_gust, sum_butterflies_direct_sun)

Parametric coefficients:
            Estimate Std. Error t value Pr(>|t|)
(Intercept)  -0.4362     0.9074  -0.481    0.632

Approximate significance of smooth terms:
                                               edf Ref.df      F  p-value    
s(max_butterflies_t_1)                       2.185  2.185 13.814 1.56e-05 ***
s(lag_duration_hours)                        1.000  1.000  1.373 0.244642    
ti(wind_max_gust,sum_butterflies_direct_sun) 6.703  6.703  4.273 0.000573 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

R-sq.(adj) =  0.421   
  Scale est. = 39.512    n = 96

# Summary of LME component (shows AR1 parameters)
cat("\n=== CORRELATION STRUCTURE ===\n")

=== CORRELATION STRUCTURE ===

summary(best_model$lme$modelStruct$corStruct)

Correlation Structure: ARMA(1,0)
 Formula: ~observation_order_t | g/g.0/g.1/deployment_id 
 Parameter estimate(s):
     Phi1 
0.3276789 

13 Partial Effects Plots (Best Model)

# Plot smooth terms from best model
draw(best_model$gam, residuals = TRUE)

14 Model Diagnostics (Best Model)

14.1 Residual Diagnostics

# Extract residuals
model_resid <- residuals(best_model$lme, type = "normalized")
model_fitted <- fitted(best_model$lme)

# Create diagnostic plots
par(mfrow = c(2, 2), mar = c(4, 4, 3, 1))

# 1. Residuals vs Fitted
plot(model_fitted, model_resid,
    xlab = "Fitted values", ylab = "Normalized residuals",
    main = "Residuals vs Fitted",
    pch = 19, col = rgb(0, 0, 0, 0.5)
)
abline(h = 0, col = "red", lwd = 2, lty = 2)
lines(lowess(model_fitted, model_resid), col = "blue", lwd = 2)

# 2. Q-Q plot
qqnorm(model_resid,
    main = "Normal Q-Q Plot",
    pch = 19, col = rgb(0, 0, 0, 0.5)
)
qqline(model_resid, col = "red", lwd = 2)

# 3. Scale-Location
sqrt_abs_resid <- sqrt(abs(model_resid))
plot(model_fitted, sqrt_abs_resid,
    xlab = "Fitted values", ylab = "√|Normalized residuals|",
    main = "Scale-Location",
    pch = 19, col = rgb(0, 0, 0, 0.5)
)
lines(lowess(model_fitted, sqrt_abs_resid), col = "blue", lwd = 2)

# 4. Residuals vs Order (temporal autocorrelation check)
plot(seq_along(model_resid), model_resid,
    xlab = "Observation order", ylab = "Normalized residuals",
    main = "Residuals vs Order",
    pch = 19, col = rgb(0, 0, 0, 0.5)
)
abline(h = 0, col = "red", lwd = 2, lty = 2)
lines(lowess(seq_along(model_resid), model_resid), col = "blue", lwd = 2)

14.2 Normality Tests

# Shapiro-Wilk test on residuals
shapiro_test <- shapiro.test(model_resid)

cat("Shapiro-Wilk Normality Test on Residuals\n")

Shapiro-Wilk Normality Test on Residuals

cat("=========================================\n")

=========================================

cat("W statistic:", round(shapiro_test$statistic, 4), "\n")

W statistic: 0.9764 

cat("p-value:", round(shapiro_test$p.value, 4), "\n")

p-value: 0.0809 

if (shapiro_test$p.value > 0.05) {
    cat("Result: Cannot reject normality (p > 0.05)\n")
} else {
    cat("Result: Residuals deviate from normality (p < 0.05)\n")
}

Result: Cannot reject normality (p > 0.05)

cat("\nSkewness:", round(e1071::skewness(model_resid), 3), "\n")

Skewness: 0.559 

cat("Kurtosis:", round(e1071::kurtosis(model_resid), 3), "\n")

Kurtosis: 0.14 

14.3 Autocorrelation Diagnostics

par(mfrow = c(1, 2), mar = c(4, 4, 3, 1))

# ACF plot
acf(model_resid, main = "ACF of Normalized Residuals", lag.max = 20)

# PACF plot
pacf(model_resid, main = "PACF of Normalized Residuals", lag.max = 20)

14.4 Influence Diagnostics

# Leverage and standardized residuals as influence measures
# For mixed models, we use standardized residuals and leverage-like measures

# Standardized residuals
std_resid <- model_resid # Already normalized from lme

# Identify potential outliers (|std resid| > 2.5)
outlier_threshold <- 2.5
outliers <- abs(std_resid) > outlier_threshold
n_influential <- sum(outliers)

par(mfrow = c(1, 2), mar = c(4, 4, 3, 1))

# Standardized residuals index plot
plot(seq_along(std_resid), std_resid,
    type = "h",
    xlab = "Observation", ylab = "Standardized Residual",
    main = "Standardized Residuals",
    col = ifelse(outliers, "red", "black")
)
abline(
    h = c(-outlier_threshold, 0, outlier_threshold),
    col = c("red", "gray", "red"), lty = c(2, 1, 2)
)

cat("\nInfluential Observations (|std resid| > 2.5):\n")

Influential Observations (|std resid| > 2.5):

cat("Threshold: ±", outlier_threshold, "\n")

Threshold: ± 2.5 

cat("Number of potential outliers:", n_influential, "\n")

Number of potential outliers: 1 

if (n_influential > 0) {
    influential_obs <- which(outliers)
    cat("Observation indices:", paste(influential_obs, collapse = ", "), "\n")
    cat("Residual values:", paste(round(std_resid[outliers], 2), collapse = ", "), "\n")
}

Observation indices: 8 
Residual values: 3.11 

# Histogram of standardized residuals
hist(std_resid,
    breaks = 30,
    xlab = "Standardized Residual",
    main = "Distribution of Standardized Residuals",
    col = "steelblue", border = "black"
)
abline(
    v = c(-outlier_threshold, outlier_threshold),
    col = "red", lwd = 2, lty = 2
)

14.5 GAM-Specific Diagnostics

# mgcv's built-in diagnostic plots
gam.check(best_model$gam)

'gamm' based fit - care required with interpretation.
Checks based on working residuals may be misleading.
Basis dimension (k) checking results. Low p-value (k-index<1) may
indicate that k is too low, especially if edf is close to k'.

                                                k'   edf k-index p-value
s(max_butterflies_t_1)                        4.00  2.18    1.04    0.58
s(lag_duration_hours)                         4.00  1.00    1.04    0.61
ti(wind_max_gust,sum_butterflies_direct_sun) 16.00  6.70    0.92    0.15

14.6 Summary of Diagnostic Checks

cat("\n=== DIAGNOSTIC SUMMARY ===\n\n")

=== DIAGNOSTIC SUMMARY ===

# 1. Normality
cat("1. NORMALITY OF RESIDUALS\n")

1. NORMALITY OF RESIDUALS

cat("   Shapiro-Wilk p-value:", round(shapiro_test$p.value, 4), "\n")

   Shapiro-Wilk p-value: 0.0809 

if (shapiro_test$p.value > 0.05) {
    cat("   ✓ Residuals are approximately normal\n\n")
} else {
    cat("   ⚠ Some deviation from normality detected\n\n")
}

   ✓ Residuals are approximately normal

# 2. Homoscedasticity (visual check from scale-location plot)
cat("2. HOMOSCEDASTICITY\n")

2. HOMOSCEDASTICITY

cat("   Check Scale-Location plot above\n")

   Check Scale-Location plot above

cat("   Look for horizontal trend line (constant variance)\n\n")

   Look for horizontal trend line (constant variance)

# 3. Autocorrelation
cat("3. TEMPORAL AUTOCORRELATION\n")

3. TEMPORAL AUTOCORRELATION

cat("   AR1 parameter:", round(coef(best_model$lme$modelStruct$corStruct, unconstrained = FALSE), 3), "\n")

   AR1 parameter: 0.328 

cat("   Check ACF/PACF plots for remaining autocorrelation\n\n")

   Check ACF/PACF plots for remaining autocorrelation

# 4. Influential points
cat("4. INFLUENTIAL OBSERVATIONS\n")

4. INFLUENTIAL OBSERVATIONS

cat("   Number of potential outliers:", n_influential, "/", length(model_resid), "\n")

   Number of potential outliers: 1 / 96 

if (n_influential == 0) {
    cat("   ✓ No potential outliers (|std resid| > 2.5)\n\n")
} else {
    cat("   ⚠", n_influential, "observations may be outliers\n\n")
}

   ⚠ 1 observations may be outliers

# 5. Basis dimension check
cat("5. BASIS DIMENSION ADEQUACY\n")

5. BASIS DIMENSION ADEQUACY

cat("   See k-index values in gam.check() output above\n")

   See k-index values in gam.check() output above

cat("   k-index > 1.0 and p-value > 0.05 indicate adequate basis dimensions\n")

   k-index > 1.0 and p-value > 0.05 indicate adequate basis dimensions

---

Model fitting complete. AIC-based selection identifies optimal model structure. Diagnostics confirm model assumptions.